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 HermiteH

 http://functions.wolfram.com/07.01.21.0008.01

 Input Form

 Integrate[(z^(\[Alpha] - 1) HermiteH[\[Nu], z])/E^(p z^2), z] == 2^\[Nu] Sqrt[Pi] ((-(z^\[Alpha]/(2 Gamma[(1 - \[Nu])/2] (p z^2)^(\[Alpha]/2)))) Sum[(Pochhammer[-(\[Nu]/2), k]/(Pochhammer[1/2, k] k! p^k)) Gamma[\[Alpha]/2 + k, p z^2], {k, 0, Infinity}] + (z^(1 + \[Alpha])/(Gamma[-(\[Nu]/2)] (p z^2)^((\[Alpha] + 1)/2))) Sum[(Pochhammer[(1 - \[Nu])/2, k]/(Pochhammer[3/2, k] k! p^k)) Gamma[(\[Alpha] + 1)/2 + k, p z^2], {k, 0, Infinity}])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p"]], " ", SuperscriptBox["z", "2"]]]], RowBox[List["HermiteH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["z", "\[Alpha]"], RowBox[List["2", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox["\[Alpha]", "2"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], SuperscriptBox["p", "k"]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], " ", "+", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], SuperscriptBox["p", "k"]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], " ", "+", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 - p z 2 H ν ( z ) z 2 ν π ( z α + 1 Γ ( - ν 2 ) ( p z 2 ) α + 1 2 k = 0 ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] Γ ( α + 1 2 + k , p z 2 ) ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! p k - z α 2 Γ ( 1 - ν 2 ) ( p z 2 ) α / 2 k = 0 ( - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] Γ ( α 2 + k , p z 2 ) ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! p k ) z z α -1 -1 p z 2 HermiteH ν z 2 ν 1 2 z α 1 Gamma -1 ν 2 -1 p z 2 α 1 2 -1 -1 k 0 Pochhammer 1 -1 ν 2 -1 k Gamma α 1 2 -1 k p z 2 Pochhammer 3 2 k k p k -1 -1 z α 2 Gamma 1 -1 ν 2 -1 p z 2 α 2 -1 -1 k 0 Pochhammer -1 ν 2 -1 k Gamma α 2 -1 k p z 2 Pochhammer 1 2 k k p k -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p_"]], " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["p", "k"]]]]]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Alpha]", "/", "2"]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], "+", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["p", "k"]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]]]]]], ")"]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29